REMOTE STORAGE 


FREE CONVECTION OF HEAT IN LIQUIDS 


BY 


ROY ANDREW NELSON 
B.S. Knox College, 1916 
M.S. University.of Illinois, 1920 


A THESIS 


SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE 
DEGREE OF DOCTOR OF PHILOSOPHY IN PHYSICS IN THE 
GRADUATE SCHOOL OF THE UNIVERSITY 
OF ILLINOIS, 1923 


Reprinted from Puysicat REvIEw, pp. 94-103, No. 1, January, 1924 








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FREE CONVECTION OF HEAT IN LIQUIDS 


By Roy A. NELSON 


ABSTRACT 


Free convection of heat from a hot wire in various liquids.—(1) Photo- 
graphic study of the convection streams. Both a parallel ray and a schlieren 
method were used and gave similar results. Photographs are reproduced for 
water, olive oil and glycerine. In all cases there is a necking down of the 
stream a few millimeters above the wire. When the wire is first heated, a 
layer of liquid all around the wire is heated; the lower half first moves down, then 
stops and goes up joining the stream which follows the upper half, which had 
proceeded as a distinct semi-circular pulse. The appearance of the image of the 
wire tends to confirm Langmuir’s suggestion of a stationary film on the surface. 
Bubbles on the wire in olive oil were observed to execute interesting oscilla- 
tions. (2) Variation of heat loss with the excess temperature of the wire over that 
of the liquid. After the wire (a silver one, .33 mm in diameter) had been heated 
electrically for about 10 minutes a steady state was reached; then the resistance 
of the wire was measured and the mean temperature of the liquid was deter- 
mined with a mercury thermometer. It was found in the case of water, 
alcohol, CCli, glycerine and castor oil, that the heat loss per cm? per sec. is equal 
to b6” when n has the values 1.15, 1.105, 1.125, 1.25 and 1.24 respectively. 
The measurements of A. H. Davis are shown to give values of m of about 1.15 
for five liquids, but there is an indication that » depends upon the diameter of 
the wire. The factor } varies with the liquid and the mean temperature and 
also perhaps with the diameter of wire. 


Part 1. PHOTOGRAPHIC STUDY 


F THE large amount of work that has been done on convection of 

heat, practically all has been from the viewpoint of the rate of loss 

from heated bodies and the effect of the physical properties of the fluid on 

this rate. Very little has been done on the motions and conditions that 
exist in the fluid due to convection currents. 

Pramanik! has made an optical study of free and forced convection 
from hot wires in air. He gives a number of photographs showing the 
convection currents, both when the currents are gravity currents and 
when a stream of air is forced past the wire. He also shows the effect of 
having several heated wires close together. 

In the present work, which was begun before Pramanik’s work ap- 
peared, a large number of photographs have been taken, showing the 
convection currents in liquids of widely different physical properties. 


1 Pramanik, Proc. Indian Assoc. Cult. Sci. 7, 115, 1922 


FREE CONVECTION OF HEAT IN LIQUIDS . 95 


1. Methods of obtaining photographs. Since the index of refraction of a 
liquid changes with the density, it is not difficult to make the convection 
currents in liquids visible and to photograph them. 

In this work two methods were used to obtain the photographs; the 
“parallel light’? method, and the “method of striae” or a modification of 
the Schlieren Method. 

The first method is shown in upper part of Fig. 1. An arc light D was 
placed at the focus of the lens ZL, so that a parallel beam of light passed 
through the box with the glass sides containing the liquid. The box was 
7.5cm by 8 cm, and 10 cm high. The heating element was a manganin 
wire 2.7 cm long and 0.0315 cm in diameter. L2and L3 were camera lenses 
and a magnification of from 3 to 4 diameters was used in obtaining the 
photographs. 




















Ree 
Fig. 1. Optical arrangements; parallel light method above; method of striae below. 

In the second method, also shown in Fig. I, a long focus lens Ls was 
placed so that a converging beam of light passed through the liquid 
along the axis of the heating wire. To shorten the distance between the 
camera and the wire so a magnification could be obtained, a lens Ls was 
used to cause the beam to converge rapidly after passing through the 
liquid. A steel hemisphere K, 0.47 cm in diameter was placed so that 
practically all the light was shut off from the camera when the liquid 
was all at the same temperature. When the heating current was estab- 
lished, the liquid in the neighborhood of the wire was heated and moved 
upward. The difference in the densities of different parts of the liquid 
caused the light to be bent around the obstacle K and an image was formed 
on the plate in the camera. 

In order to compare the two methods for showing the convection 
streams, photographs were taken of convection streams under almost the 
same conditions. The photograph in Fig. 2a was taken by the parallel 
light method and Fig. 2b by the method of striae, the magnifications 
being about 5 times and 3.5 times respectively. The same general char- 
acteristics may be seen in both photographs. The parallel light method 


96 ROY A. NELSON 





s#Pigures 2,3 ana 4. Water. 





2a. Parallel light. 2b. Method of striae. 





Fig. 5. Olive oil 





Sa. Kh sm 1.2: Sb. h 


i 
" 


Es oo. Db 6.9- 





4a. after 3 sec.; 4b. 1 sec.; 4c. 13 sec. Fig. 





Convection streams in water, olive oil and glycerine 
Fig. 2. Comparison of two photographic methods; water 
Fig. 3. Variation with heat loss h, in cal. per cm? per sec. 
Fig. 4. Variation with time after starting the current. 


FREE CONVECTION OF HEAT IN LIQUIDS 97 


shows the convection streams at a greater distance above the wire because 
a larger region was illuminated in using this method. Photographs taken 
using the schlieren method were found to show more details of the convec- 
tion stream near the wire and so this method was used when the convec- 
tion flow near the wire was studied. The stream of rising fluid acts as a 
lens for the beam of light passing through it. This was shown by changing 
the position of the lens LZ, in Fig. 1. The convection stream could be 
made to appear bright as shown in the photographs or, if the lens Lz was 
moved farther from the box the image would be dark with a bright edge. 

If the glass box was turned so that the light passed through the liquid 
perpendicular to the axis of the wire, it was found that the image of the 
wire could be made much larger when the heating current was on than 
when it was off. If the lens ZL, was moved toward the wire a position 
could be found where the image of the wire practically disappeared when 
the heating current was turned on. This spreading out of image of the 
wire seemed to be the same above the wire as it was below. This is what 
would happen if Langmuir’s theory that there is a stationary film on a 
heated surface is true for liquids. The spreading of the image would not 
be the same below the wire as it is above if the liquid film was not sta- 
tionary but was being constantly replaced by unheated liquid. Some of 
the photographs which will be discussed later on also tend to confirm this 
theory. 

2. Convection currents in water. In water it was found that it was 
necessary with the size wire used to have a rate of heat loss of at least 
0.8 calories per cm? per sec. in order that convection streams would be 
visible. Figs. 3a, 3b, and 3c show photographs taken of water having 
heat losses of 1.28, 3.71, 6.86 cal per cm? per sec., respectively. The 
distances between the horizontal lines of the photographs represent 
distances of one centimeter. The photographs show that at a short dis- 
tance above the wire the width of the convection streams is the largest. 
This is no doubt due to the heated liquid coming out from below the wire. 
But at a short distance farther up the stream narrows down and is very 
sharp. Then as it proceeds up it begins to diffuse out into the cooler 
liquid. It is interesting to note that there is a region in which the flow is 
practically linear. Figs. 3b and 3c show this quite distinctly. 

Figs. 4a, 4b, 4c show the formation of the convection stream. These 
photographs were taken, using the method of striae. The heat loss in 
these four cases was all the same, 6.7 cal per cm? per sec, but they were 
taken at 4, 1 and 1.5 sec., respectively, after the current was turned on. 
Fig. 4a shows the heated layer close to the wire and the distance moved 
upward is very small. From visual observation, the beginning of the 


98 ROY A. NELSON 


convection flow had the appearance of a wave moving outward from the 
wire, but when the lower half had proceeded a short distance down it was 
reflected and moved upward. Fig. 4b shows this wave moving up. The 
central dark spot represents the position of the wire. Fig. 4c shows the 
wave which proceeds upward. The wave could be seen until it reached 
the surface of the water. These photographs all seem to substantiate 
Langmuir’s film theory. If the molecules of the liquid received heat only 
as they came in contact or very close to the wire, the heated liquid would 
not be seen at the lower side of the wire. 

If the box was turned so the light passed through the liquid at right 
angles to the axis of the wire and the current was turned on, the wave 
shown in the previous photographs appeared as a “wave front”? moving 
upward. Over the central portion of the wire this ‘‘wave front” was 
straight unless there was some disturbing condition such as bubbles or 
particles clinging to the wire. Above the ends of the wire the end effect 
could be seen and the “wave front”? was seen to curve downward. 

3. Convection currents in alcohol. Using alcohol as the convecting 
liquid, the same general characteristics were observed. The stream does 
not converge as sharply as in the case of water but convection currents for 
a lower rate of heat loss could be made visible. The velocity of the initial 
wave which travels up is higher than in water because of the lower vis- 
cosity. It was noticed in alcohol and not in the other liquids tried, that 
near the surface the bright image had a tendency to move slowly back 
and forth, even when the liquid as a whole seemed perfectly quiet. 

4. Convection currents in olive oil and glycerine. Fig. 5 shows the 
stream lines in olive oil after the heat current had been on for fifteen 
minutes and a steady state had been reached. When the heating current 
was first turned on a wave of the same nature as shown in Fig. 4 moved 
slowly upward. 

Fig. 6 shows the same effect in glycerine. This photograph was taken 
25 minutes after the heating current was turned on. 

An interesting phenomenon was observed in olive oil with the light 
passing through the box perpendicular to the wire. When the wire was 
heated so that bubbles formed on the wire, these bubbles would move 
back and forth along the wire. If there was only one bubble it would 
vibrate over nearly the entire length of the wire. If there was more than 
one bubble, they would oscillate back and forth, rebounding when two 
came close to each other. After some time the bubble would come to 
rest. When the heating current was turned off the bubbles would rise 
to the surface. This same phenomena was observed in glycerine but the 
oscillatory motion was damped out much quicker. 


FREE CONVECTION OF HEAT IN LIQUIDS 99 


Another point of interest in this connection was that if bubbles were 
formed when the heating current was turned on, and if after the wave 
like that shown in Fig. 4c had proceeded a centimeter or two above the 
wire the switch was opened, the bubbles would rise rapidly until they 
reached this wave and then rise to the surface with it, but would not pass 
through the wave. 


PART I]. MEASUREMENTS OF HEAT LOSSES FROM A WIRE 


The first experimental study of free convection seems to have been made 
by Dulong and Petit.?, They studied the rate of loss of heat from ther- 
mometer bulbs in gases maintained at a constant temperature. They found 
that the rate of loss of heat by convection was proportional to 6!?%3 where 
6 represents the excess in temperature of the heated body over that of 
the gas. 

Lorenz’ solved mathematically the case for the rate of heat loss from a 
vertical plate by making certain assumptions regarding the air currents 






pecaachee 


Big. :/ 


near the plate. He found that the rate of heat loss should be proportional 
to 6/4. This has been verified by experiments on small vertical plates 
by Langmuir,‘ and on large vertical plates at the National Physical 
Laboratory of Great Britain.* Langmuir‘ has also shown that for heated 
wires in gases the same exponent of @ holds for temperatures up to 600°C. 

The present investigation was undertaken to determine whether the 
variation of convection with the temperature excess for liquids follows a 
simple law such as that for gases: h=b6". 

1. Experimental methods. The apparatus used is shown diagrammati- 
cally in Fig. 7. The heating wire was of silver and was 0.033 cm in 


* Dulong and Petit, Ann. de Chim. et Phys. 7, 1817 

3 Lorenz, Ann. der Phys. 13, 582, 1881 

4 Langmuir, Trans. Amer. Electrochem. Soc. 23, 299, 1913 

§ Report on ‘‘Heat Transmission by Convection and Radiation” published by the 
Food Investigation Board. 

6 Langmuir, Phys. Rev. 34, 401, 1912 


100 ROY A. NELSON 


diameter and 16.9 cm in length. The liquid under investigation was 
contained in a glass box 25 cm by 10 cm by 12 cm. The temperature of 
the wire was determined by measurements of its resistance, using a 
Kelvin Double Bridge. The potential drop leads were silver wires 0.015 
cm in diameter and soldered to the heating wire 2.2 cm from the ends. 
The temperature coefficient of the heating wire was found to be 0.0388. 
The potential drop over the 0.01 ohm resistance S was compared with 
that of the silver wire. The heating current was taken from a set of 
storage batteries having a large capacity, the current thus being main- 
tained amply constant. 

From preliminary experiments it was found that the temperature of 
the wire increased rather rapidly when the heating current was first 
turned on, while after the current had run for about 10 minutes the 
change in temperaturé of the wire was proportional to the time and 
corresponded to the rise in temperature of the liquid as a whole. In 
taking the readings the heating current was allowed to run until this 
steady state had been reached; then the resistance was measured and 
the liquid stirred; then the mean temperature of the liquid was deter- 
mined with a mercury thermometer. 

2. Results of experiments. The liquids used in the experiments were 
water, alcohol, carbon tetrachloride, glycerine and castor oil. These 
liquids differ greatly in their physical properties and especially in regard 
to viscosity. In Table I the constants for these liquids are given. The 
values given in the table were principally taken from the 7th Edition of 
the Smithsonian Physical Tables and from Landolt-Bornstein “Phy- 
sikalisch-Chemisch Tabellen.”’ 


TABLE I 
Physical constants of liquids studied 

Liquid p n C R10)" a (10)4 
Alcohol 0.792 0.011 0.46 4.3 110 
Toluene 0.866 0.005 0.34 pal 10.9 
Aniline 025 0.045 0.53 4.1 Sao 
CCl, 1.582 0.0097 eos 225 2.66 
Water 1.000 0.0101 1.00 14.0 3.0 
Olive oil 0.916 0.989 0.43 O59 BPA 
Glycerine i226 8.5 0.73 Ta) 6.8 
Castor oil 0.969 | 0.42 4.2 
po =density n = viscosity c=specific heat per unit vol. 
k =thermal expansion a=coefficient of cubical expansion 


In Table II the results of a representative set of readings for water are 
given. The current through the wire and the resistance are given in the 
first two columns, the heat loss in calories per cm? per sec. /, is given in 
the third column. 6 is the temperature difference between the wire and 


FREE CONVECTION OF HEAT IN LIQUIDS 101 


the liquid, i.e.,9=t#y—t,. The value of the exponent 7 in the equation 
h = b6" was obtained by plotting the logarithms of / and @ and determining 
the slope of the lines. The ratio 1/6” is given in the last column. 


TABLE II 
Water Mean temperature 24.8°C n=1.147 
I R h tw tL 6 gn h/on 


4.91 .026030 0.1151 26.28 24.36 1.94 2.143 .0537 
8.33 .026312 0.3354 29.30 23.34 4.96 6.276 .0535 


12.85 .026866 0.8147 35.34 24.46 10.88 15.45 .0528 
15.86 .027397 1.266 41.10 24.86 16.24 24.47 .0517 
20.88 .028392 AME ay! DAN the 26.19 42.33 .0537 


26.40 . 029946 3.833 68.94 21,9] 40.97 70.71 .0542 


The results obtained for the different liquids used are collected in 
Table III. 


TABLE IIIT 
Liquid t n h/on Liquid t n h/on 
water. IV eh GE Da bo eet .0422 CCh, istGew (else Olos 
water 19 1.165 .0458 CCl 22 1.117 .0144 
water 24 1.147 F525 glycerine 19 1.244 .0099 
water 25 1.147 .0533 glycerine 2 fF Sul es ee 
water Ze 1.141 .0582 castor oil 6752-51-26 .00442 
alcohol 20 1.105 50212 castor oil 18307 1921 .00601 
mreono 92025 ~)-8.105 .0216 castor oil S420 e127 .00589 


A. H. Davis’ gives values for the relation between the rate of heat loss 
and the temperature difference. He used a Wheatstone bridge method 
of measuring the resistance of the heated wire and by compensating leads 
corrected for the end effects. He assumed that the heat loss per unit 
length was proportional to 6? but his table shows that the ratio H/6 
was far from constant. 

In Table IV the results given by Davis have been recalculated by the 
present writer and the value of the exponent m determined for each 
liquid. The heat loss is expressed in calories per cm? per sec. instead of 
calories per cm length per sec. as in his table. 

The results show that the values of the exponent should be between 1 
and 1.25 instead of the value 7 =2 assumed by Davis. 

From the present writers’ experiments, therefore, and from the present 
examination of Davis’ measurements, it appears that the rate of loss of 
heat is not proportional to 6? but to 6” where m has a value somewhat 
greater than unity. For all liquids tried except glycerine and castor oil, 
the value of is not far from 1.15. In using liquids of high viscosity it 
was very difficult to obtain steady readings and in plotting the graph 


7A. H. Davis, Phil. Mag. 44, 933, 1922 


102 ROY A. NELSON 


from the logarithms of f and 6, it was found that the points were more 
scattered for glycerine and castor oil than for the other liquids. Davis 
states that his results for glycerine were the least satisfactory. This may 
account for the difference between the values of 1 for glycerine obtained 
from Davis’ results and from those of the present writer. 


‘TABLESLY. 


Values of h/0", where h=heat loss in cal/cm and 0 =excess temperature 
(Calculated by the present writer from A. H. Davis’ measurements) 


wire 


Liquid diameter n 1 2 5 10 20 50 
Toluene 0.0083 ica Mial .0305 .0305 .0308 .0305 .0307 
0.0155 1.154 .0202 .0198 .0194 .0196 .0199 .0204 
CCl 0.0083 1.127 .0242 .0238 .0241 .0240 .0241 .0240 
0.0155 1.157 .0166 .0168 .0163 .0167 (0166) 90%eG 
Aniline 0.0083 1.173 .0274 .0286 .0277 .0278 [027608 g02ee 
0.0155 1.167 .0205 .0201 .0196 .0197 .0202 .0207 
Glycerine 0.0083 1.000 .0368 .0360 .0368 .0395 
0.0155 ias2 .0183 .0172 .0171 .0176. (O83 
Olive oil 0.0083 1.146 .0184 .0185 .0183 .0183 .0186 
0.0155 ivis3 .0124 .0118 .0120 .0121 .0125 


In Tables III and IV the values of the ratio 4/6” are not constant but 
vary with the liquid, the mean temperature of the liquid and probably 
with the size of the heating wire. However, no definite conclusion can 
be drawn from the change in the ratio h/6" with the wire diameter from 
Table IV, because the mean temperatures of the liquids were not given 
by Davis. The value of the exponent 7 is nearly the same for the different 
liquids and does not change much when the temperature of the liquid is 
changed. There is an indication from Table IV that the value of 
changes with the size of the heating wire, but the accuracy of the meas- 
urements is not sufficient to show definitely whether or not this is true. 
Since is nearly the same for liquids of widely different physical prop- 
erties, it is evident that the value of the coefficient 6 must be the pre- 
dominating factor in determining the free convection of heat by a liquid. 

It is planned to carry out further investigation on the loss of heat by 
convection in other liquids and determine the effect of the size of heating 
wire on the value of the coefficient 6 and on the value of the exponent 7. ' 
If it is found that the value of the coefficient b is independent of the diam- 
eter of the wire, then this coefficient may be taken as a measure of the 
convective cooling power of aliquid. Even if 6 is a function of the diam- 
eter of the wire, the relative cooling power of different liquids may be 
found by comparing the values of 6 when determined with the same size 


FREE CONVECTION OF HEAT IN LIQUIDS 103 


heating wire. It may also be possible to express b as function of the 
physical properties of the liquid but so far no simple relationship has 
been found. 

Rayleigh® suggests the possibility of applying the principle of similitude 
to the problem of convection of heat. Richardson’ and Davis!® have 
applied this method and obtained relations for the rate of loss of heat 
from similar bodies immersed in viscous fluids. Davis has shown that the 
grouping of variables given by the principle of similitude gives satis- 
factory results for convection in gas, carbon tetrachloride, aniline, toulene, 
olive oil and glycerine. He also shows that the thermal conductivity of 
the liquid should be the predominating physical property in facilitating 
cooling by convection. The values of the ratio h/6" given in Table III 
bear out this statement, since water has the highest thermal conductivity 
and also gives the largest value for the ratio h/6". 

In a recent paper, Rice! has applied Langmuir’s film theory and dimen- 
sional reasoning to obtain equations for free and forced convection in a 
liquid. He has used free convection as a method of determining the 
thermal conductivities of toluene, glycerine, olive oil, aniline, carbon 
tetrachloride and No. 12 transil oil. 

The writer wishes to acknowledge his indebtedness to Professor 
A. P. Carman for the interest in and the suggestions made in the course 
of this investigation. — 


LABORATORY OF PHYSICS, 
UNIVERSITY OF ILLINOIS, 
URBANA, ILLINOIS, 
June, 1923.!° 


8 Rayleigh, Nature, 45, 66, 1915 

9 Richardson, Proc. Phys. Soc., May 1920 

10 Davis, Phil. Mag. 40, 692, 1920; 44, 329, and 928, 1922 
11 Rice, (Abstract) Phys. Rev. 21, 474, Apr. 1923 

12 Received July 14, 1923 


VITA 


Roy ANDREW NELSON, born February 15, 1896, near Joy, Illinois. 
Graduated from Knox College with the degree of Bachelor of Science 
in 1916. Graduate Student at the University of Illinois, September, 
1916 to November, 1917, and September, 1919 to June, 1923, receiving 
the degree of Master of Science, June, 1920. In the United States 
Army from April, 1918 to January, 1919. Completed Officers’ Train- 
ing Course at Fort Monroe in September, 1918, and commissioned 2nd 
Lieutenant in Coast Artillery. Assistant in Physics, September, 1916 
to November, 1917, and September, 1919 to June, 1923. Taught in the 
Summer Sessions 1917 and 1921, Publications: Bulletin No. 122 En- 
gineering Experiment Station, ‘‘Thermal Conductivity and Diffusivity 
of Concrete” by A. P. Carman and R. A. Nelson. Physical Review, 
Ser. 2, Vol. 18, p. 113, “Thermal Conductivity of White Marble and 
‘Neat’ Cement.” 








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